On the correlation structure of closed queueing networks

Closed networks of exponential queues are prototypes of systems which show strong negative dependence properties. We use the device of a test customer traveling in the network to evaluate the qualitative behavior of the network's correlation. The results are: negative association of successive sojourn times during a cycle; negative correlation of two successive cycle times; feedback-sojourn times are positively associated; correlation between cycles vanishes over long distances geometrically fast. Generalizations to networks with general topology are given. Additionally we prove for cyclic networks that the conditional cycle time stochastically increases when the initial joint queue length vector increases with respect to the partial sum ordering. This connects the space-time correlation with space-time monotonicity.